Mathematical Sciences: Crystalline Variational Problems

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Jean Taylor will continue her work on surfaces of minimum energy. She will focus her attention on minimizing energy functionals which do not possess directional uniformity, this is known as anisotropism. The techniques to be used come from a branch of the calculus of variations, geometric measure theory, which is having an ever increasing impact on many areas of geometric analysis. This work has applications in materials research and some joint projects with John Cahn, a metallurgist, will be continued. One of her main objectives will be to understand the geometric consequences of minimization of crystal- type anisotropic surface energies. In a slightly different direction she will try to develop and employ combinatorial techniques of area minimization. These latter techniques become appropriate in the polyhedral situations which frequently arise. The energy functionals have an integral representation inwhich the integrand, known as the free energy, is defined on the unit sphere. The free energy in turn gives rise to the Wulff shape, which is the obect of most of Taylor's studies. She will study several new types of singularities in these surfaces and some new rules for orientations in heterogeneous nucleation, which have been predicted in some of her earlier work with Cahn. Some of this will involve allowing edge energies in the integral being minimized. Other investigations will study the evolution of the minimizing surfaces as the energy function is allowed to vary.